Properties

Label 2-384-8.5-c3-0-17
Degree $2$
Conductor $384$
Sign $i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 2.82i·5-s − 14.1·7-s − 9·9-s + 20i·11-s − 39.5i·13-s − 8.48·15-s − 34·17-s − 52i·19-s − 42.4i·21-s − 62.2·23-s + 117·25-s − 27i·27-s − 200. i·29-s − 110.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.252i·5-s − 0.763·7-s − 0.333·9-s + 0.548i·11-s − 0.844i·13-s − 0.146·15-s − 0.485·17-s − 0.627i·19-s − 0.440i·21-s − 0.564·23-s + 0.936·25-s − 0.192i·27-s − 1.28i·29-s − 0.639·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7366792226\)
\(L(\frac12)\) \(\approx\) \(0.7366792226\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3iT \)
good5 \( 1 - 2.82iT - 125T^{2} \)
7 \( 1 + 14.1T + 343T^{2} \)
11 \( 1 - 20iT - 1.33e3T^{2} \)
13 \( 1 + 39.5iT - 2.19e3T^{2} \)
17 \( 1 + 34T + 4.91e3T^{2} \)
19 \( 1 + 52iT - 6.85e3T^{2} \)
23 \( 1 + 62.2T + 1.21e4T^{2} \)
29 \( 1 + 200. iT - 2.43e4T^{2} \)
31 \( 1 + 110.T + 2.97e4T^{2} \)
37 \( 1 + 271. iT - 5.06e4T^{2} \)
41 \( 1 - 26T + 6.89e4T^{2} \)
43 \( 1 - 252iT - 7.95e4T^{2} \)
47 \( 1 - 345.T + 1.03e5T^{2} \)
53 \( 1 + 681. iT - 1.48e5T^{2} \)
59 \( 1 - 364iT - 2.05e5T^{2} \)
61 \( 1 + 735. iT - 2.26e5T^{2} \)
67 \( 1 + 628iT - 3.00e5T^{2} \)
71 \( 1 - 333.T + 3.57e5T^{2} \)
73 \( 1 + 338T + 3.89e5T^{2} \)
79 \( 1 + 789.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3iT - 5.71e5T^{2} \)
89 \( 1 + 234T + 7.04e5T^{2} \)
97 \( 1 + 178T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.59383195248158608611394665281, −9.804494897635004849793359926636, −9.059853747151346150785586980783, −7.899503691105174176646368595144, −6.83248560405468646779664952422, −5.85020192010675186257053136414, −4.69010627665298541652862564958, −3.55137003746182330282803491242, −2.42598263830314613612738146260, −0.25486677170884438806167587978, 1.35427127267376563876911138928, 2.83489369265530440013514325255, 4.08496260060414511643724764386, 5.49283355215879112799614352490, 6.48318552662670867163721831558, 7.23753796132891260030602296272, 8.505622132753173292373583581897, 9.125580676017375478451013815640, 10.25839028405504696884475520191, 11.21274302896840530641706007652

Graph of the $Z$-function along the critical line